Simplify square root of 8a^7b^5* square root of 2a^3b^3

$\sqrt{8 a^{7} b^{5}} \cdot \sqrt{2 a^{3} b^{3}}$

Step 1

Rewrite $8 a^{7} b^{5}$ as ${(2 a^{3} b^{2})}^{2} \cdot (2 a b)$ .

Factor $4$ out of $8$ .

$\sqrt{4 (2) a^{7} b^{5}} \cdot \sqrt{2 a^{3} b^{3}}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 2 a^{7} b^{5}} \cdot \sqrt{2 a^{3} b^{3}}$

Factor out $a^{6}$ .

$\sqrt{2^{2} \cdot 2 (a^{6} a) b^{5}} \cdot \sqrt{2 a^{3} b^{3}}$

Rewrite $a^{6}$ as ${(a^{3})}^{2}$ .

$\sqrt{2^{2} \cdot 2 ({(a^{3})}^{2} a) b^{5}} \cdot \sqrt{2 a^{3} b^{3}}$

Factor out $b^{4}$ .

$\sqrt{2^{2} \cdot 2 ({(a^{3})}^{2} a) (b^{4} b)} \cdot \sqrt{2 a^{3} b^{3}}$

Rewrite $b^{4}$ as ${(b^{2})}^{2}$ .

$\sqrt{2^{2} \cdot 2 ({(a^{3})}^{2} a) ({(b^{2})}^{2} b)} \cdot \sqrt{2 a^{3} b^{3}}$

Move $a$ .

$\sqrt{2^{2} \cdot 2 ({(a^{3})}^{2}) {(b^{2})}^{2} a b} \cdot \sqrt{2 a^{3} b^{3}}$

Move $2$ .

$\sqrt{(2^{2} ({(a^{3})}^{2})) {(b^{2})}^{2} \cdot 2 a b} \cdot \sqrt{2 a^{3} b^{3}}$

Rewrite $(2^{2} ({(a^{3})}^{2})) {(b^{2})}^{2}$ as ${(2 a^{3} b^{2})}^{2}$ .

$\sqrt{{(2 a^{3} b^{2})}^{2} \cdot 2 a b} \cdot \sqrt{2 a^{3} b^{3}}$

Add parentheses.

$\sqrt{{(2 a^{3} b^{2})}^{2} \cdot 2 (a b)} \cdot \sqrt{2 a^{3} b^{3}}$

Add parentheses.

$\sqrt{{(2 a^{3} b^{2})}^{2} \cdot (2 a b)} \cdot \sqrt{2 a^{3} b^{3}}$

Step 2

Pull terms out from under the radical.

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{2 a^{3} b^{3}}$

Step 3

Rewrite $2 a^{3} b^{3}$ as ${(a b)}^{2} \cdot (2 a b)$ .

Factor out $a^{2}$ .

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{2 (a^{2} a) b^{3}}$

Factor out $b^{2}$ .

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{2 (a^{2} a) (b^{2} b)}$

Move $a$ .

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{2 (a^{2}) b^{2} a b}$

Move $2$ .

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{(a^{2}) b^{2} \cdot 2 a b}$

Rewrite $(a^{2}) b^{2}$ as ${(a b)}^{2}$ .

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{{(a b)}^{2} \cdot 2 a b}$

Add parentheses.

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{{(a b)}^{2} \cdot 2 (a b)}$

Add parentheses.

$2 a^{3} b^{2} \sqrt{2 a b} \cdot \sqrt{{(a b)}^{2} \cdot (2 a b)}$

Step 4

Pull terms out from under the radical.

$2 a^{3} b^{2} \sqrt{2 a b} \cdot (a b \sqrt{2 a b})$

Step 5

Multiply $a^{3}$ by $a$ by adding the exponents.

Move $a$ .

$2 (a \cdot a^{3}) b^{2} \sqrt{2 a b} \cdot (b \sqrt{2 a b})$

Multiply $a$ by $a^{3}$ .

Raise $a$ to the power of $1$ .

$2 (a^{1} a^{3}) b^{2} \sqrt{2 a b} \cdot (b \sqrt{2 a b})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 a^{1 + 3} b^{2} \sqrt{2 a b} \cdot (b \sqrt{2 a b})$

Add $1$ and $3$ .

$2 a^{4} b^{2} \sqrt{2 a b} \cdot (b \sqrt{2 a b})$

Step 6

Multiply $b^{2}$ by $b$ by adding the exponents.

Move $b$ .

$2 a^{4} (b \cdot b^{2}) \sqrt{2 a b} \cdot \sqrt{2 a b}$

Multiply $b$ by $b^{2}$ .

Raise $b$ to the power of $1$ .

$2 a^{4} (b^{1} b^{2}) \sqrt{2 a b} \cdot \sqrt{2 a b}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 a^{4} b^{1 + 2} \sqrt{2 a b} \cdot \sqrt{2 a b}$

Add $1$ and $2$ .

$2 a^{4} b^{3} \sqrt{2 a b} \cdot \sqrt{2 a b}$

Step 7

Multiply $2 a^{4} b^{3} \sqrt{2 a b} \sqrt{2 a b}$ .

Raise $\sqrt{2 a b}$ to the power of $1$ .

$2 a^{4} b^{3} ({\sqrt{2 a b}}^{1} \sqrt{2 a b})$

Raise $\sqrt{2 a b}$ to the power of $1$ .

$2 a^{4} b^{3} ({\sqrt{2 a b}}^{1} {\sqrt{2 a b}}^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 a^{4} b^{3} {\sqrt{2 a b}}^{1 + 1}$

Add $1$ and $1$ .

$2 a^{4} b^{3} {\sqrt{2 a b}}^{2}$

Step 8

Rewrite ${\sqrt{2 a b}}^{2}$ as $2 a b$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 a b}$ as ${(2 a b)}^{\frac{1}{2}}$ .

$2 a^{4} b^{3} {({(2 a b)}^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 a^{4} b^{3} {(2 a b)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$2 a^{4} b^{3} {(2 a b)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 a^{4} b^{3} {(2 a b)}^{\frac{2}{2}}$

Rewrite the expression.

$2 a^{4} b^{3} {(2 a b)}^{1}$

Simplify.

$2 a^{4} b^{3} (2 a b)$

Step 9

Multiply $a^{4}$ by $a$ by adding the exponents.

Move $a$ .

$2 (a \cdot a^{4}) b^{3} (2 b)$

Multiply $a$ by $a^{4}$ .

Raise $a$ to the power of $1$ .

$2 (a^{1} a^{4}) b^{3} (2 b)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 a^{1 + 4} b^{3} (2 b)$

Add $1$ and $4$ .

$2 a^{5} b^{3} (2 b)$

Step 10

Multiply $b^{3}$ by $b$ by adding the exponents.

Move $b$ .

$2 a^{5} (b \cdot b^{3}) \cdot 2$

Multiply $b$ by $b^{3}$ .

Raise $b$ to the power of $1$ .

$2 a^{5} (b^{1} b^{3}) \cdot 2$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 a^{5} b^{1 + 3} \cdot 2$

Add $1$ and $3$ .

$2 a^{5} b^{4} \cdot 2$

Step 11

Multiply $2$ by $2$ .

$4 a^{5} b^{4}$

Simplify square root of 8a^7b^5* square root of 2a^3b^3

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